The component sizes of a critical random graph with given degree sequence

TL;DR

This paper analyzes the asymptotic distribution of component sizes in a critical random graph with a given degree sequence, revealing different scaling limits depending on the degree distribution's moments.

Contribution

It establishes the scaling limits of component sizes in critical random graphs with specified degree distributions, including finite third moment and power-law cases.

Findings

Component sizes scaled by n^{-2/3} converge to Brownian excursion lengths.

Component sizes with power-law degree distribution converge to excursions of a drifted process.

Results apply to critical simple graphs with finite third moment degree distribution.

Abstract

Consider a critical random multigraph $\mathcal{G}_n$ with $n$ vertices constructed by the configuration model such that its vertex degrees are independent random variables with the same distribution $\nu$ (criticality means that the second moment of $\nu$ is finite and equals twice its first moment). We specify the scaling limits of the ordered sequence of component sizes of $\mathcal{G}_n$ as $n$ tends to infinity in different cases. When $\nu$ has finite third moment, the components sizes rescaled by $n^{-2/3}$ converge to the excursion lengths of a Brownian motion with parabolic drift above past minima, whereas when $\nu$ is a power law distribution with exponent $\gamma\in(3,4)$, the components sizes rescaled by $n^{-(\gamma -2)/(\gamma-1)}$ converge to the excursion lengths of a certain nontrivial drifted process with independent increments above past minima. We deduce the asymptotic behavior of the component sizes of a critical random simple graph when $\nu$ has finite third moment.